The present invention relates to wavefront analysis generally and to various applications of wavefront analysis.
The following patents and publications are believed to represent the current state of the art:
U.S. patents:
U.S. Pat. Nos. 5,969,855; 5,969,853; 5,936,253; 5,870,191; 5,814,815; 5,751,475; 5,619,372; 5,600,440; 5,471,303; 5,446,540; 5,235,587; 4,407,569; 4,190,366;
Non-U.S. patents:
JP 9230247 (Abstract); JP 9179029 (Abstract); JP 8094936 (Abstract); JP 7261089 (Abstract); JP 7225341 (Abstract); JP 6186504 (Abstract);
Other publications:
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Creath K. xe2x80x9cPhase measurement interferometry techniquesxe2x80x9dxe2x80x94Progress in Optics XXVI, 348 (1988).
Greivenkamp J. E. xe2x80x9cGeneralized data reduction for heterodyne interferometryxe2x80x9dxe2x80x94Optical Engineering, Vol. 23, 350 (1984).
Morgan C. J. xe2x80x9cLeast-squares estimation in phase-measurement interferometryxe2x80x9dxe2x80x94Optics Letters, Vol. 7, 368 (1982).
Golden L. J. xe2x80x9cZernike test. 1: Analytical aspectsxe2x80x9dxe2x80x94Applied Optics, Vol. 16, 205 (1977).
Bruning J. H. et al. xe2x80x9cDigital wavefront measuring interferometer for testing optical surfaces and lensesxe2x80x9dxe2x80x94Applied Optics, Vol. 13, 2693 (1974).
The present invention seeks to provide methodologies and systems for wavefront analysis as well as for surface mapping, phase change analysis, spectral analysis, object inspection, stored data retrieval, three-dimensional; imaging and other suitable applications utilizing wavefront analysis.
There is thus provided in accordance with a preferred embodiment of the present invention a method of wavefront analysis. The method includes obtaining a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed which has an amplitude and a phase, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed.
There is also provided in accordance with a preferred embodiment of the present an apparatus for wavefront analysis including a wavefront transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed which has an amplitude and a phase, an intensity map generator operating to provide a plurality of intensity maps of the plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps for providing an output indicating the amplitude and phase of the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the plurality of intensity maps are employed to provide an analytical output indicating the amplitude and phase.
Still further in accordance with a preferred embodiment of the present invention the plurality of differently phase changed transformed wavefronts are obtained by interference of the wavefront being analyzed along a common optical path.
Additionally in accordance with a preferred embodiment of the present invention the plurality of differently phase changed transformed wavefronts are realized in a manner substantially different from performing a delta-function phase change to the transformed wavefront.
Further in accordance with a preferred embodiment of the present invention the plurality of intensity maps are employed to obtain an output indicating the phase which is substantially free from halo and shading off distortions.
Preferably, the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts resulting from at least one of application of spatial phase changes to a transformed wavefront and transforming of a wavefront following application of spatial phase changes thereto.
Additionally in accordance with a preferred embodiment of the present invention, the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to the wavefront being analyzed thereby to obtain a transformed wavefront and applying a plurality of different phase changes to the transformed wavefront, thereby to obtain a plurality of differently phase changed transformed wavefronts. Preferably, the plurality of different phase changes includes spatial phase changes and the plurality of different spatial phase changes are effected by applying a time-varying spatial phase change to part of the transformed wavefront.
Further in accordance with a preferred embodiment of the present invention the plurality of different spatial phase changes are effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront. Preferably, the transform applied to the wavefront being analyzed is a Fourier transform and wherein the step of obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts includes applying a Fourier transform to the plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the wavefront being analyzed is a Fourier transform and the plurality of different spatial phase changes includes at least three different phase changes. Preferably, the plurality of intensity maps includes at least three intensity maps and the step of employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed includes: expressing the wavefront being analyzed as a first complex function which has an amplitude and phase identical to the amplitude and phase of the wavefront being analyzed, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex function, having an absolute value and a phase, as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change. Expressing each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference between the phase of the wavefront being analyzed and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function and obtaining the phase of the wavefront being analyzed by adding the phase of the second complex function to the difference between the phase of the wavefront being analyzed and the phase of the second complex function.
Further in accordance with a preferred embodiment of the present invention the absolute value of the second complex function is obtained by approximating the absolute value to a polynomial of a given degree.
Still further in accordance with a preferred embodiment of the present invention the second complex function is obtained by expressing the second complex function as an eigen-value problem where the complex function is an eigen-vector obtained by an iterative process.
Preferably the second complex function is obtained by: approximating the Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change to a polynomial and approximating the second complex function to a polynomial.
Preferably, the wavefront being analyzed, the absolute value of the second complex function, and the difference between the phase of the second complex function and the phase of the wavefront being analyzed, are obtained by a least-square method, which has increased accuracy as the number of the plurality of intensity maps increases.
Further in accordance with a preferred embodiment of the present invention the plurality of different phase changes includes at least four different phase changes, the plurality of intensity maps includes at least four intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed and includes: expressing each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference between the phase of the wavefront being analyzed and the phase of the second complex function, a known phase delay produced by one of the at least four different phase changes in which each corresponds to one of the at least four intensity maps and at least one additional unknown relating to the wavefront analysis, where the number of the at least one additional unknown is no greater than the number by which the plurality intensity maps exceeds three and solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function, the difference between the phase of the wavefront being analyzed and the phase of the second complex function and the additional unknown.
Further in accordance with a preferred embodiment of the present invention the phase changes are chosen as to maximize contrast in the intensity maps and to minimize effects of noise on the phase of the wavefront being analyzed.
Preferably, expressing each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference between the phase of the wavefront being analyzed and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes which corresponds to one of the at least three intensity maps and includes: defining fourth, fifth and sixth complex functions, none of which being a function of any of the plurality of intensity maps or of the time-varying spatial phase change, each of the fourth, fifth and sixth complex functions being a function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function and expressing each of the plurality of intensity maps as a sum of the fourth complex function, the fifth complex function multiplied by the sine of the known phase delay corresponding to each one of the plurality of intensity maps and the sixth complex function multiplied by the cosine of the known phase delay corresponding to each one of the plurality of intensity maps.
Preferably, the step of solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function includes: obtaining two solutions for each of the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, the two solutions being a higher value solution and a lower value solution, combining the two solutions into an enhanced absolute value solution for the absolute value of the second complex function, by choosing at each spatial location either the higher value solution or the lower value solution of the two solutions in a way that the enhanced absolute value solution satisfies the second complex function. Preferably, combining the two solutions of the amplitude of the wavefront being analyzed into enhanced amplitude solution, by choosing at each spatial location the higher value solution or the lower value solution of the two solutions of the amplitude in the way that at each location where the higher value solution is chosen for the absolute value solution, the higher value solution is chosen for the amplitude solution and at each location where the lower value solution is chosen for the absolute value solution, the lower value solution is chosen for the amplitude solution, combining the two solutions of the difference between the phase of the wavefront being analyzed and the phase of the second complex function into an enhanced difference solution, by choosing at each spatial location the higher value solution or the lower value solution of the two solutions of the difference in the way that at each location where the higher value solution is chosen for the absolute value solution, the higher value solution is chosen for the difference solution and at each location where the lower value solution is chosen for the absolute value solution, the lower value solution is chosen for the difference solution.
Further in accordance with a preferred embodiment of the present invention the spatially uniform, time-varying spatial phase change is applied to a spatially central part of the transformed wavefront.
Preferably, the transform applied to the wavefront being analyzed is a Fourier transform and wherein the step of obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts includes applying a Fourier transform to the plurality of differently phase changed transformed wavefronts.
Still further in accordance with a preferred embodiment of the present invention the method also includes adding a phase component including relatively high frequency components to the wavefront being analyzed prior to applying the transform thereto in order to increase the high-frequency content of the transformed wavefront prior to the applying the spatially uniform, time-varying spatial phase change to part of the transformed wavefront.
Preferably, the spatially uniform, time-varying spatial phase change is applied to a spatially centered generally circular region of the transformed wavefront and the spatially uniform, time-varying spatial phase change is applied to approximately one half of the transformed wavefront.
Additionally in accordance with a preferred embodiment of the present invention the transformed wavefront includes a DC region and a non-DC region and the spatially uniform, time-varying spatial phase change is applied to at least part of both the DC region and the non-DC region.
Further in accordance with a preferred embodiment of the present invention the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been changed by employing an at least time varying phase change function. Alternatively, the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been changed by applying an at least time varying phase change function to the wavefront being analyzed.
Preferably, the at least time varying phase change function is applied to the wavefront being analyzed prior to transforming thereof. Alternatively, the at least time varying phase change function is applied to the wavefront being analyzed subsequent to transforming thereof.
Further in accordance with a preferred embodiment of the present invention the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been changed by employing an at least time varying phase change function.
Additionally or alternatively, the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been changed by applying an at least time varying phase change function to the wavefront to be analyzed.
Preferably, the at least time varying phase change function is a spatially uniform spatial function.
Additionally in accordance with a preferred embodiment of the present invention the transformed wavefront includes a plurality of different wavelength components and the plurality of different spatial phase changes are effected by applying a phase change to the plurality of different wavelength components of the transformed wavefront.
Preferably, the phase change applied to the plurality of different wavelength components of the transformed wavefront is a time-varying spatial phase change.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components of the transformed wavefront is effected by passing the transformed wavefront through an object, at least one of whose thickness and refractive index varies spatially.
Still further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components of the transformed wavefront is effected by reflecting the transformed wavefront from a spatially varying surface.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components of the transformed wavefront is selected to be different to a predetermined extent for at least some of the plurality of different wavelength components.
Additionally in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components of the transformed wavefront is identical for at least some of the plurality of different wavelength components.
Further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes a plurality of different wavelength components. Preferably, the plurality of differently phase changed transformed wavefronts are obtained by applying a phase change to the plurality of different wavelength components of the wavefront being analyzed.
Preferably, the phase change is applied to the plurality of different wavelength components of the wavefront being analyzed prior to transforming thereof.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components is effected by passing the wavefront being analyzed through an object, at least one of whose thickness and refractive index varies spatially.
Further in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of intensity maps is performed simultaneously for all of the plurality of different wavelength components and obtaining a plurality of intensity maps includes dividing the plurality of phase changed transformed wavefronts into separate wavelength components.
Still further in accordance with a preferred embodiment of the present invention the step of dividing the plurality of phase changed transformed wavefronts is effected by passing the plurality of phase changed transformed wavefronts through a dispersion element.
Additionally in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components is effected by passing the wavefront being analyzed through an object, at least one of whose thickness and refractive index varies spatially, following transforming of the wavefront being analyzed.
Preferably, the phase change which is applied to the plurality of different wavelength components is effected by reflecting the wavefront being analyzed from a spatially varying surface, following transforming of the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components is selected to be different to a predetermined extent for at least some of the plurality of different wavelength components. Preferably, the phase change which is applied to the plurality of different wavelength components is identical for at least some of the plurality of different wavelength components.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different wavelength components is effected by passing the wavefront being analyzed through a plurality of objects, each characterized in that at least one of its thickness and refractive index varies spatially.
Preferably, the phase change applied to the plurality of different wavelength components is effected by passing the wavefront being analyzed through a plurality of objects, each characterized in that at least one of its thickness and refractive index varies spatially, following transforming of the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes a plurality of different polarization components and the plurality of differently phase changed transformed wavefronts are obtained by applying a phase change to the plurality of different polarization components of the wavefront being analyzed prior to transforming thereof.
Still further in accordance with a preferred embodiment of the present invention the transformed wavefront includes a plurality of different polarization components and the plurality of different spatial phase changes are effected by applying a phase change to the plurality of different polarization components of the transformed wavefront.
Additionally in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different polarization components of the transformed wavefront is different for at least some of the plurality of different polarization components.
Further in accordance with a preferred embodiment of the present invention the phase change applied to the plurality of different polarization components of the transformed wavefront is identical for at least some of the plurality of different polarization components.
Additionally in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts includes applying a transform to the plurality of differently phase changed transformed wavefronts.
Preferably, the plurality of phase changed transformed wavefronts are reflected from a reflecting surface so that the transform applied to the plurality of differently phase changed transformed wavefronts is identical to the transform applied to the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the transform applied to the wavefront being analyzed is a Fourier transform.
Still further in accordance with a preferred embodiment of the present invention the plurality of intensity maps are obtained by reflecting the plurality of differently phase changed transformed wavefronts from a reflecting surface so as to transform the plurality of differently phase changed transformed wavefronts.
Additionally in accordance with a preferred embodiment of the present invention the method of obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts includes applying a transform to the plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the method of employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed includes expressing the plurality of intensity maps as at least one mathematical function of phase and amplitude of the wavefront being analyzed and employing the at least one mathematical function to obtain an output indicating the phase and amplitude.
Preferably, the method of employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed includes expressing the plurality of intensity maps as at least one mathematical function of phase and amplitude of the wavefront being analyzed and of the plurality of different phase changes, wherein the phase and amplitude are unknowns and the plurality of different phase changes are known and employing the at least one mathematical function to obtain an output indicating the phase and amplitude.
Further in accordance with a preferred embodiment of the present invention the plurality of intensity maps includes at least four intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed, includes employing a plurality of combinations, each of at least three of the plurality of intensity maps, to provide a plurality of indications of the amplitude and phase of the wavefront being analyzed.
Preferably, the method also includes employing the plurality of indications of the amplitude and phase of the wavefront being analyzed to provide an enhanced indication of the amplitude and phase of the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention at least some of the plurality of indications of the amplitude and phase are at least second order indications of the amplitude and phase of the wavefront being analyzed.
Further in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to the wavefront being analyzed, thereby obtaining a transformed wavefront and applying a plurality of different phase and amplitude changes to the transformed wavefront, thereby obtaining a plurality of differently phase and amplitude changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the plurality of different phase and amplitude changes includes at least three different phase and intensity changes, the plurality of different phase and amplitude changes are effected by applying at least one of a spatially uniform, time-varying spatial phase change and a spatially uniform, time-varying spatial amplitude change to at least part of the transformed wavefront, the plurality of intensity maps includes at least three intensity maps. Preferably, the step of employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed includes: expressing the wavefront being analyzed as a first complex function which has an amplitude and phase identical to the amplitude and phase of the wavefront being analyzed, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing at least one of a spatially uniform, time-varying spatial phase change and a spatially uniform, time-varying spatial amplitude change, defining a second complex function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change, expressing each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function and a difference between the phase of the wavefront being analyzed and the phase of the second complex function. Preferably, the spatial function governing at least one of a spatially uniform, time-varying spatial phase change and a spatially uniform, time-varying spatial amplitude change includes: defining fourth, fifth, sixth and seventh complex functions, none of which is a function of any of the plurality of intensity maps or of the time-varying spatial phase change. Preferably, each of the fourth, fifth, sixth and seventh complex functions being a function of at least one of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, defining an eighth function of a phase delay and of an amplitude change, both produced by one of the at least three different phase and amplitude changes, corresponding to the at least three intensity maps and expressing each of the plurality of intensity maps as a sum of the fourth complex function, the fifth complex function multiplied by the absolute value squared of the eighth function, the sixth complex function multiplied by the eighth function and the seventh complex function multiplied by the complex conjugate of the eighth function, solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function and obtaining the phase of the wavefront being analyzed by adding the phase of the second complex function to the difference between the phase of the wavefront being analyzed and phase of the second complex function.
Further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes at least two wavelength components. Preferably, the step of obtaining a plurality of intensity maps also includes dividing the phase changed transformed wavefronts according to the at least two wavelength components in order to obtain at least two wavelength components of the phase changed transformed wavefronts and in order to obtain at least two sets of intensity maps, each set corresponding to a different one of the at least two wavelength components of the phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the wavefront being analyzed, obtaining an output indicative of the phase of the wavefront being analyzed from each of the at least two sets of intensity maps and combining the outputs to provide an enhanced indication of phase of the wavefront being analyzed, in which enhanced indication, there is no 2xcfx80 ambiguity.
Additionally in accordance with a preferred embodiment of the present invention the wavefront being analyzed is an acoustic radiation wavefront.
Still further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes at least one one-dimensional component, the transform applied to the wavefront being analyzed is a one-dimensional Fourier transform, performed in a dimension perpendicular to a direction of propagation of the wavefront being analyzed, thereby to obtain at least one one-dimensional component of the transformed wavefront in the dimension perpendicular to the direction of propagation. The plurality of differently phase changed transformed wavefronts are obtained by applying the plurality of different phase changes to each of the at least one one-dimensional component, thereby obtaining at least one one-dimensional component of the plurality of phase changed transformed wavefronts and the plurality of intensity maps are employed to obtain an output indicating amplitude and phase of the at least one one-dimensional component of the wavefront being analyzed.
Preferably, the plurality of different phase changes is applied to each of the one-dimensional component by providing a relative movement between the wavefront being analyzed and an element. Preferably, the element generates spatially varying, time-constant phase changes, the relative movement being in an additional dimension which is perpendicular both to the direction of propagation and to the dimension perpendicular to the direction of propagation.
Further in accordance with a preferred embodiment of the present invention the wavefront being analyzed includes a plurality of different wavelength components, the plurality of different phase changes are applied to the plurality of different wavelength components of each of the plurality of one-dimensional components of the wavefront being analyzed and the step of obtaining a plurality of intensity maps includes dividing the plurality of one-dimensional components of the plurality of phase changed transformed wavefronts into separate wavelength components.
Still further in accordance with a preferred embodiment of the present invention dividing the plurality of one-dimensional components of the plurality of phase changed transformed wavefronts into separate wavelength components is achieved by passing the plurality of phase changed transformed wavefronts through a dispersion element.
Further in accordance with a preferred embodiment of the present invention the transform applied to the wavefront being analyzed includes an additional Fourier transform to minimize cross-talk between different one-dimensional components of the wavefront being analyzed.
There is provided in accordance with another preferred embodiment of the present invention a method of surface mapping. The method includes obtaining a surface mapping wavefront having an amplitude and a phase, by reflecting radiation from a surface and analyzing the surface mapping wavefront by: obtaining a plurality of differently phase changed transformed wavefronts corresponding to the surface mapping wavefront, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the surface mapping wavefront.
There is further provided in accordance with a preferred embodiment of the present invention an apparatus for surface mapping. The apparatus includes a wavefront obtainer operating to obtain a surface mapping wavefront having an amplitude and a phase, by reflecting radiation from a surface, a wavefront analyzer, analyzing the surface mapping wavefront and including a wavefront transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to the surface mapping wavefront, an intensity map generator operating to provide a plurality of intensity maps of the plurality of phase changed transformed wavefronts and an intensity map utilizer, the plurality of intensity maps provide an output indicating the amplitude and phase of the surface mapping wavefront.
Further in accordance with a preferred embodiment of the present invention the radiation reflected from the surface has a narrow band about a given wavelength, causing the phase of the surface mapping wavefront to be proportional to geometrical variations in the surface, the proportion being an inverse linear function of the wavelength.
Still further in accordance with a preferred embodiment of the present invention the radiation reflected from the surface has at least two narrow bands, each centered about a different wavelength, providing at least two wavelength components in the surface mapping wavefront and at least two indications of the phase of the surface mapping wavefront, thereby enabling an enhanced mapping of the surface to be obtained by avoiding an ambiguity in the mapping which exceeds the larger of the different wavelengths about which the two narrow bands are centered.
Additionally in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to the surface mapping wavefront, thereby to obtain a transformed wavefront and applying a plurality of different phase changes, including spatial phase changes, to the transformed wavefront, thereby to obtain a plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the surface mapping wavefront is a Fourier transform, the plurality of different phase changes includes at least three different phase changes, effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront, the plurality of intensity maps includes at least three intensity maps. Preferably, the step of employing the plurality of intensity maps to obtain an output indicates the amplitude and phase of the surface mapping wavefront and includes expressing the surface mapping wavefront as a first complex function which has an amplitude and phase identical to the amplitude and phase of the surface mapping wavefront, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change. Preferably, expressing each of the plurality of intensity maps as a third function of: the amplitude of the surface mapping wavefront, the absolute value of the second complex function, a difference between the phase of the surface mapping wavefront and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes, corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the surface mapping wavefront, the absolute value of the second complex function and the difference between the phase of the surface mapping wavefront and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function and obtaining the phase of the surface mapping wavefront by adding the phase of the second complex function to the difference between the phase of the surface mapping wavefront and phase of the second complex function.
Preferably, the surface mapping wavefront includes a plurality of different wavelength components. The plurality of differently phase changed transformed wavefronts are preferably obtained by: transforming the surface mapping wavefront thereby obtaining a transformed wavefront including a plurality of different wavelength components and applying a phase change to the plurality of different wavelength components of the transformed wavefront by passing the transformed wavefront through an object, at least one of whose thickness and refractive index varies spatially.
There is also provided in accordance with yet another preferred embodiment of the present invention a method of inspecting an object. The method includes obtaining an object inspection wavefront which has an amplitude and a phase, by transmitting radiation through the object and analyzing the object inspection wavefront by: obtaining a plurality of differently phase changed transformed wavefronts corresponding to the object inspection wavefront, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the object inspection wavefront.
There is further provided in accordance with a preferred embodiment of the present invention an apparatus for inspecting an object. The apparatus includes a wavefront obtainer operating to obtain an object inspection wavefront which has an amplitude and a phase, by transmitting radiation through the object, a wavefront analyzer, analyzing the object inspection wavefront and including a wavefront transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to the object inspection wavefront, an intensity map generator operating to provide a plurality of intensity maps of the plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to provide an output indicating the amplitude and phase of the object inspection wavefront.
Preferably, when the object is substantially uniform in material and other optical properties, the phase of the object inspection wavefront is proportional to the object thickness.
Additionally, when the object is substantially uniform in thickness, the phase of the object inspection wavefront is proportional to optical properties of the object.
Further in accordance with a preferred embodiment of the present invention the radiation has at least two narrow bands, each centered about a different wavelength, providing at least two wavelength components in the object inspection wavefront and at least two indications of the phase of the object inspection wavefront, thereby enabling an enhanced mapping of thickness of the object to be inspected by avoiding an ambiguity in the mapping which exceeds the larger of the different wavelengths about which the two narrow bands are centered.
Still further in accordance with a preferred embodiment of the present invention the method of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to the object inspection wavefront, thereby obtaining a transformed wavefront and applying a plurality of different phase changes, including spatial phase changes, to the transformed wavefront, thereby obtaining a plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the object inspection wavefront is a Fourier transform, the plurality of different phase changes includes at least three different phase changes, effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront. Preferably, the plurality of intensity maps includes at least three intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the object inspection wavefront and includes: expressing the object inspection wavefront as a first complex function which has an amplitude and phase identical to the amplitude and phase of the object inspection wavefront, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change, expressing each of the plurality of intensity maps as a third function of: the amplitude of the object inspection wavefront, the absolute value of the second complex function, a difference between the phase of the object inspection wavefront and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes, corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the object inspection wavefront, the absolute value of the second complex function and the difference between the phase of the object inspection wavefront and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function and obtaining the phase of the object inspection wavefront by adding the phase of the second complex function to the difference between the phase of the object inspection wavefront and phase of the second complex function.
Still further in accordance with a preferred embodiment of the present invention the object inspection wavefront includes a plurality of different wavelength components. The plurality of differently phase changed transformed wavefronts are preferably obtained by: transforming the object inspection wavefront thereby obtaining a transformed wavefront including a plurality of different wavelength components and applying a phase change to the plurality of different wavelength components of the transformed wavefront by reflecting the transformed wavefront from a spatially varying surface.
There is also provided in accordance with yet another preferred embodiment of the present invention a method of spectral analysis. The method includes obtaining a spectral analysis wavefront having an amplitude and a phase, by causing radiation to impinge on an object, analyzing the spectral analysis wavefront by: obtaining a plurality of differently phase changed transformed wavefronts corresponding to the spectral analysis wavefront which has an amplitude and a phase, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the spectral analysis wavefront and employing the output indicating the amplitude and phase to obtain an output indicating spectral content of the radiation.
There is provided in accordance with a further preferred embodiment of the present invention an apparatus for spectral analysis. The apparatus includes a wavefront obtainer operating to obtain a spectral analysis wavefront having an amplitude and a phase, by causing radiation to impinge on an object, a wavefront analyzer, analyzing the spectral analysis wavefront, including a wavefront transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to the spectral analysis wavefront which has an amplitude and a phase, an intensity map generator operating to provide a plurality of intensity maps of the plurality of phase changed transformed wavefronts, an intensity map utilizer, employing the plurality of intensity maps to provide an output indicating the amplitude and phase of the spectral analysis wavefront and a phase and amplitude utilizer, employing the output indicating the amplitude and phase to obtain an output indicating spectral content of the radiation.
Further in accordance with a preferred embodiment of the present invention and wherein obtaining the spectral analysis wavefront is effected by reflecting the radiation from the object.
Still further in accordance with a preferred embodiment of the present invention and wherein obtaining the spectral analysis wavefront is effected by transmitting the radiation through the object.
Additionally in accordance with a preferred embodiment of the present invention the radiation is substantially of a single wavelength, the phase of the spectral analysis wavefront is inversely proportional to the single wavelength, and is related to at least one of a surface characteristic and thickness of the impinged object.
Still further in accordance with a preferred embodiment of the present invention the step of employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the spectral analysis wavefront includes: expressing the plurality of intensity maps as at least one mathematical function of phase and amplitude of the spectral analysis wavefront and of the plurality of different phase changes, wherein at least the phase is unknown and a function generating the plurality of phase changed transformed wavefronts is known and employing the at least one mathematical function to obtain an output indicating at least the phase.
Additionally in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to the spectral analysis wavefront, thereby obtaining a transformed wavefront and applying a plurality of different phase changes, including spatial phase changes, to the transformed wavefront, thereby obtaining a plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the spectral analysis wavefront is a Fourier transform, the plurality of different phase changes includes at least three different phase changes, effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront. Preferably, the plurality of intensity maps includes at least three intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the spectral analysis wavefront includes: expressing the spectral analysis wavefront as a first complex function which has an amplitude and phase identical to the amplitude and phase of the spectral analysis wavefront, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change, expressing each of the plurality of intensity maps as a third function of: the amplitude of the spectral analysis wavefront, the absolute value of the second complex function, a difference between the phase of the spectral analysis wavefront and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes, corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the spectral analysis wavefront, the absolute value of the second complex function and the difference between the phase of the spectral analysis wavefront and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function and obtaining the phase of the spectral analysis wavefront by adding the phase of the second complex function to the difference between the phase of the spectral analysis wavefront and phase of the second complex function.
Further in accordance with a preferred embodiment of the present invention the spectral analysis wavefront includes a plurality of different wavelength components and the plurality of differently phase changed transformed wavefronts are obtained by applying a phase change to the plurality of different wavelength components of the spectral analysis wavefront.
There is further provided in accordance with a preferred embodiment of the present invention a method of phase change analysis. The method includes obtaining a phase change analysis wavefront which has an amplitude and a phase, applying a transform to the phase change analysis wavefront thereby to obtain a transformed wavefront, applying a plurality of different phase changes to the transformed wavefront, thereby to obtain a plurality of differently phase changed transformed wavefronts, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output indication of differences between the plurality of different phase changes applied to the transformed phase change analysis wavefront.
There is also provided in accordance with yet another preferred embodiment of the present invention an apparatus for phase change analysis. The apparatus includes a wavefront obtainer, operating to obtain a phase change analysis wavefront which has an amplitude and a phase, a transform applier, applying a transform to the phase change analysis wavefront thereby to obtain a transformed wavefront, a phase change applier, applying at least one phase change to the transformed wavefront, thereby to obtain at least one phase changed transformed wavefront, an intensity map generator operating to provide at least one intensity map of the phase changed transformed wavefront and an intensity map utilizer, employing the plurality of intensity maps to provide an output indication of differences between the plurality of different phase changes applied to the transformed phase change analysis wavefront.
Typically, when lateral shifts appear in the plurality of different phase changes, corresponding changes appear in the plurality of intensity maps and the step of employing the plurality of intensity maps results in obtaining an indication of the lateral shifts.
Still further in accordance with a preferred embodiment of the present invention the step of employing the plurality of intensity maps to obtain an output indication of differences between the plurality of different phase changes applied to the transformed phase change analysis wavefront includes: expressing the plurality of intensity maps as at least one mathematical function of phase and amplitude of the phase change analysis wavefront and of the plurality of different phase changes, where at least the phase and amplitude are known and the plurality of different phase changes are unknown and employing the mathematical function to obtain an output indicating the differences between the plurality of different phase changes.
There is further provided in accordance with yet a further preferred embodiment of the present invention a method of phase change analysis. The method includes obtaining a phase change analysis wavefront which has an amplitude and a phase, applying a transform to the phase change analysis wavefront thereby to obtain a transformed wavefront, applying at least one phase change to the transformed wavefront, thereby to obtain at least one phase changed transformed wavefront, obtaining at least one intensity map of the at least one phase changed transformed wavefront and employing the intensity map to obtain an output indication of the at least one phase change applied to the transformed phase change analysis wavefront.
There is also provided in accordance with yet another preferred embodiment of the present invention an apparatus for phase change analysis. The apparatus includes a wavefront obtainer, operating to obtain a phase change analysis wavefront which has an amplitude and a phase, a transform applier, applying a transform to the phase change analysis wavefront thereby to obtain a transformed wavefront, a phase change applier, applying at least one phase change to the transformed wavefront, thereby to obtain at least one phase changed transformed wavefront, an intensity map generator operating to provide at least one intensity map of the phase changed transformed wavefront and an intensity map utilizer, employing the intensity map to provide an output indication of the phase change applied to the transformed phase change analysis wavefront.
Preferably, the phase change is a phase delay, having a value selected from a plurality of pre-determined values, and the output indication of the phase change includes the value of the phase delay.
There is also provided in accordance with a preferred embodiment of the present invention a method of stored data retrieval. The method includes obtaining a stored data retrieval wavefront which has an amplitude and a phase, by reflecting radiation from the media in which information is encoded, by selecting the height of the media at each of a multiplicity of different locations on the media. Preferably, analyzing the stored data retrieval wavefront by: obtaining a plurality of differently phase changed transformed wavefronts corresponding to the stored data retrieval wavefront, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an indication of the amplitude and phase of the stored data retrieval wavefront and employing the indication of the amplitude and phase to obtain the information.
There is further provided in accordance with yet another preferred embodiment of the present invention an apparatus for stored data retrieval. The apparatus includes a wavefront obtainer operating to obtain a stored data retrieval wavefront which has an amplitude and a phase, by reflecting radiation from the media in which information is encoded by selecting the height of the media at each of a multiplicity of different locations on the media, a wavefront analyzer, analyzing the stored data retrieval wavefront and including a wavefront transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to the stored data retrieval wavefront, an intensity map generator operating to obtain a plurality of intensity maps of the plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to provide an indication of the amplitude and phase of the stored data retrieval wavefront and a phase and amplitude utilizer employing the indication of the amplitude and phase to provide the information.
Preferably, the step of obtaining a plurality of differently phase changed transformed wavefronts includes: applying a transform to the stored data retrieval wavefront thereby to obtain a transformed wavefront and applying a plurality of different phase changes to the transformed wavefront, thereby to obtain a plurality of differently phase changed transformed wavefronts.
Further in accordance with a preferred embodiment of the present invention the transform applied to the stored data retrieval wavefront is a Fourier transform, the plurality of different phase changes includes at least three different phase changes, effected by applying a spatially uniform, time-varying spatial phase change to part of the transformed wavefront, the plurality of intensity maps includes at least three intensity maps and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the stored data retrieval wavefront includes: expressing the stored data retrieval wavefront as a first complex function which has an amplitude and phase identical to the amplitude and phase of the stored data retrieval wavefront, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change, expressing each of the plurality of intensity maps as a third function of: the amplitude of the stored data retrieval wavefront, the absolute value of the second complex function, a difference between the phase of the stored data retrieval wavefront and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes, corresponding to one of the at least three intensity maps, solving the third function to obtain the amplitude of the stored data retrieval wavefront, the absolute value of the second complex function and the difference between the phase of the stored data retrieval wavefront and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function and obtaining the phase of the stored data retrieval wavefront by adding the phase of the second complex function to the difference between the phase of the stored data retrieval wavefront and phase of the second complex function.
Still further in accordance with a preferred embodiment of the present invention the stored data retrieval wavefront includes at least one one-dimensional component, the transform applied to the data retrieval wavefront is a one-dimensional Fourier transform, performed in a dimension perpendicular to a direction of propagation of the data retrieval wavefront, thereby to obtain at least one one-dimensional component of the transformed wavefront in the dimension perpendicular to the direction of propagation, the plurality of differently phase changed transformed wavefronts are obtained by applying the plurality of different phase changes to each of the one-dimensional component, thereby obtaining at least one one-dimensional component of the plurality of phase changed transformed wavefronts and the plurality of intensity maps are employed to obtain an output indicating amplitude and phase of the one-dimensional component of the data retrieval wavefront.
Preferably, the plurality of different phase changes is applied to each of the at least one one-dimensional component by providing a relative movement between the media and a component generating spatially varying, time-constant phase changes, the relative movement being in a dimension perpendicular to the direction of propagation and to the dimension perpendicular to the direction of propagation.
Additionally in accordance with a preferred embodiment of the present invention the information is encoded on the media whereby: an intensity value is realized by reflection of light from each location on the media to lie within a predetermined range of values, the range corresponding an element of the information stored at the location and by employing the plurality of intensity maps, multiple intensity values are realized for each location, providing multiple elements of information for each location on the media.
Preferably, the plurality of differently phase changed transformed wavefronts include a plurality of wavefronts whose phase has been changed by applying an at least time varying phase change function to the stored data retrieval wavefront.
Further in accordance with a preferred embodiment of the present invention the stored data retrieval wavefront includes a plurality of different wavelength components and the plurality of differently phase changed transformed wavefronts are obtained by applying at least one phase change to the plurality of different wavelength components of the stored data retrieval wavefront.
Further in accordance with a preferred embodiment of the present invention the radiation which is reflected from the media includes a plurality of different wavelength components, resulting in the stored data retrieval wavefront including a plurality of different wavelength components and the plurality of differently phase changed transformed wavefronts are obtained by applying a phase change to the plurality of different wavelength components of the stored data retrieval wavefront.
Still further in accordance with a preferred embodiment of the present invention the information encoded by selecting the height of the media at each of a multiplicity of different locations on the media is also encoded by selecting the reflectivity of the media at each of a plurality of different locations on the media and employing the indication of the amplitude and phase to obtain the information includes employing the indication of the phase to obtain the information encoded by selecting the height of the media and employing the indication of the amplitude to obtain the information encoded by selecting the reflectivity of the media.
There is provided in accordance with another preferred embodiment of the present invention a method of 3-dimensional imaging. The method includes obtaining a 3-dimensional imaging wavefront, which has an amplitude and a phase, by reflecting radiation from an object to be viewed and analyzing the 3-dimensional imaging wavefront by: obtaining a plurality of differently phase changed transformed wavefronts corresponding to the 3-dimensional imaging wavefront, obtaining a plurality of intensity maps of the plurality of differently phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output indicating the amplitude and phase of the 3-dimensional imaging wavefront.
There is further provided in accordance with a preferred embodiment of the present invention an apparatus for 3-dimensional imaging. The apparatus includes a wavefront obtainer operating to obtain a 3-dimensional imaging wavefront, which has an amplitude and a phase, by reflecting radiation from an object to be viewed, a wavefront analyzer, analyzing the 3-dimensional imaging wavefront including a wavefront transformer operative to provide a plurality of differently phase changed transformed wavefronts corresponding to the 3-dimensional imaging wavefront, an intensity map generator operative to provide a plurality of intensity maps of the plurality of differently phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to provide an output indicating the amplitude and phase of the 3-dimensional imaging wavefront.
Further in accordance with a preferred embodiment of the present invention the radiation reflected from the object has a narrow band about a given wavelength, causing the phase of the 3-dimensional imaging wavefront to be proportional to geometrical variations in the object, the proportion being an inverse linear function of the wavelength.
Additionally in accordance with a preferred embodiment of the present invention the step of obtaining a plurality of differently phase changed transformed wavefronts includes applying a transform to the 3-dimensional imaging wavefront, thereby to obtain a transformed wavefront and applying a plurality of different phase changes, including spatial phase changes, to the transformed wavefront, thereby to obtain a plurality of differently phase changed transformed wavefronts.
Still further in accordance with a preferred embodiment of the present invention the 3-dimensional imaging wavefront includes a plurality of different wavelength components and the plurality of differently phase changed transformed wavefronts are obtained by: transforming the 3-dimensional imaging wavefront, thereby obtaining a transformed wavefront including a plurality of different wavelength components and applying phase changes to the plurality of different wavelength components of the transformed wavefront by passing the transformed wavefront through an object, at least one of whose thickness and refractive index varies spatially.
There is also provided in accordance with yet another preferred embodiment of the present invention a method of wavefront analysis. The method includes obtaining a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to obtain an output indicating at least the phase of the wavefront being analyzed by combining the plurality of intensity maps into a second plurality of combined intensity maps, the second plurality being less than the first plurality, obtaining at least an output indicative of the phase of the wavefront being analyzed from each of the second plurality of combined intensity maps and combining the outputs to provide at least an enhanced indication of phase of the wavefront being analyzed.
There is also provided in accordance with yet another preferred embodiment of the present invention an apparatus wavefront analysis. The apparatus includes a wavefront transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed, an intensity map generator operating to obtain a plurality of intensity maps of the plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to obtain an output indicating at least amplitude of the wavefront being analyzed and including an intensity combiner operating to combine the plurality of intensity maps into a second plurality of combined intensity maps, the second plurality being less than the first plurality, an indication provider operating to provide at least an output indicative of the amplitude of the wavefront being analyzed from each of the second plurality of combined intensity maps and an enhanced indication provider, combining the outputs to provide at least an enhanced indication of amplitude of the wavefront being analyzed.
There is provided in accordance with a further preferred embodiment of the present invention a method of wavefront analysis. The method includes obtaining a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefront and employing the plurality of intensity maps to obtain an output indicating at least amplitude of the wavefront being analyzed by combining the plurality of intensity maps into a second plurality of combined intensity maps, the second plurality being less than the first plurality, obtaining at least an output indicative of the amplitude of the wavefront being analyzed from each of the second plurality of combined intensity maps and combining the outputs to provide at least an enhanced indication of amplitude of the wavefront being analyzed.
There is provided in accordance with a preferred embodiment of the present invention an apparatus for wavefront analysis. The apparatus includes a wavefront transformer operating to provide a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed, an intensity map generator operating to provide a plurality of intensity maps of the plurality of phase changed transformed wavefronts and an intensity map utilizer, employing the plurality of intensity maps to provide an output indicating at least the phase of the wavefront being analyzed. Preferably, the apparatus also includes an intensity map expresser, expressing the plurality of intensity maps as a function of: amplitude of the wavefront being analyzed, phase of the wavefront being analyzed and a phase change function characterizing the plurality of differently phase changed transformed wavefronts, a complex function definer, defining a complex function of: the amplitude of the wavefront being analyzed, the phase of the wavefront being analyzed and the phase change function characterizing the plurality of differently phase changed transformed wavefronts, the complex function being characterized in that the intensity at each location in the plurality of intensity maps is a function predominantly of a value of the complex function at the location and of the amplitude and the phase of the wavefront being analyzed at the location. The apparatus also typically, includes complex function expresser, expressing the complex function as a function of the plurality of intensity maps and a phase obtainer, obtaining values for the phase by employing the complex function expressed as a function of the plurality of intensity maps.
There is also provided in accordance with another preferred embodiment of the present invention a method of wavefront analysis. The method includes obtaining a plurality of differently phase changed transformed wavefronts corresponding to a wavefront being analyzed, obtaining a plurality of intensity maps of the plurality of phase changed transformed wavefronts and employing the plurality of intensity maps to provide an output indicating at least the phase of the wavefront being analyzed by: expressing the plurality of intensity maps as a function of: amplitude of the wavefront being analyzed, phase of the wavefront being analyzed and a phase change function characterizing the plurality of differently phase changed transformed wavefronts. Additionally, defining a complex function of: the amplitude of the wavefront being analyzed, the phase of the wavefront being analyzed and the phase change function characterizing the plurality of differently phase changed transformed wavefronts, the complex function being characterized in that the intensity at each location in the plurality of intensity maps is a function predominantly of a value of the complex function at the location and of the amplitude and the phase of the wavefront being analyzed at the location, expressing the complex function as a function of the plurality of intensity maps and obtaining values for the phase by employing the complex function expressed as a function of the plurality of intensity maps.
There is further provided in accordance with yet a further preferred embodiment of the present invention a method of wavefront analysis. The method includes applying a Fourier transform to a wavefront being analyzed which has an amplitude and a phase, thereby obtaining a transformed wavefront, applying a spatially uniform time-varying spatial phase change to part of the transformed wavefront, thereby to obtain at least three differently phase changed transformed wavefronts, applying a second Fourier transform to obtain at least three intensity maps of the at least three phase changed transformed wavefronts and employing the at least three intensity maps to obtain an output indicating at least one of the phase and the amplitude of the wavefront being analyzed by: expressing the wavefront being analyzed as a first complex function which has an amplitude and phase identical to the amplitude and phase of the wavefront being analyzed, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change, defining a second complex function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change, expressing each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference between the phase of the wavefront being analyzed and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes, which each correspond to one of the at least three intensity maps, solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, solving the second complex function to obtain the phase of the second complex function and obtaining the phase of the wavefront being analyzed by adding the phase of the second complex function to the difference between the phase of the wavefront being analyzed and phase of the second complex function.
There is further provided in accordance with yet a further preferred embodiment of the present invention an apparatus for wavefront analysis. The apparatus includes a first transform applier, applying a Fourier transform to a wavefront being analyzed which has an amplitude and a phase thereby to obtain a transformed wavefront, a phase change applier, applying a spatially uniform time-varying spatial phase change to part of the transformed wavefront, thereby obtaining at least three differently phase changed transformed wavefronts, a second transform applier, applying a second Fourier transform to the at least three differently phase changed transformed wavefronts, thereby obtaining at least three intensity maps. The apparatus also typically includes an intensity map utilizer, employing the at least three intensity maps to provide an output indicating the phase and the amplitude of the wavefront being analyzed and a wavefront expresser, expressing the wavefront being analyzed as a first complex function which has an amplitude and phase identical to the amplitude and phase of the wavefront being analyzed, a first intensity map expresser, expressing the plurality of intensity maps as a function of the first complex function and of a spatial function governing the spatially uniform, time-varying spatial phase change. Preferably, the apparatus also includes a complex function definer, defining a second complex function having an absolute value and a phase as a convolution of the first complex function and of a Fourier transform of the spatial function governing the spatially uniform, time-varying spatial phase change, a second intensity map expresser, expressing each of the plurality of intensity maps as a third function of: the amplitude of the wavefront being analyzed, the absolute value of the second complex function, a difference between the phase of the wavefront being analyzed and the phase of the second complex function and a known phase delay produced by one of the at least three different phase changes, which each correspond to one of the at least three intensity maps. The apparatus further typically includes a first function solver, solving the third function to obtain the amplitude of the wavefront being analyzed, the absolute value of the second complex function and the difference between the phase of the wavefront being analyzed and the phase of the second complex function, a second function solver, solving the second complex function to obtain the phase of the second complex function and a phase obtainer, obtaining the phase of the wavefront being analyzed by adding the phase of the second complex function to the difference between the phase of the wavefront being analyzed and the phase of the second complex function.